Onset of Bénard–Marangoni ferroconvection with a convective surface boundary condition: The effects of cubic temperature profile and MFD viscosity

The combined effects of basic cubic temperature profiles and magnetic field dependent (MFD) viscosity on the onset of Bénard-Marangoni convection in a ferrofluid layer are studied. The lower boundary is rigid-isothermal, while the upper free boundary open to the atmosphere is flat and subject to a general thermal boundary condition. The Galerkin technique is employed to extract the critical stability parameters numerically. The results indicate that the basic cubic temperature profiles have a profound influence on the stability characteristics of the system and can be effectively used to either suppress or augment the onset of Bénard–Marangoni ferroconvection. Besides, increasing the magnetic Rayleigh number and the nonlinearity of magnetization hastens, while an increase in the Biot number and MFD viscosity parameter delays the onset of Bénard–Marangoni ferroconvection. The existing results in the literature are obtained as particular cases from the present study.     

Effect of magnetic field dependent viscosity and rotation on ferroconvection saturating a porous medium in the presence of dust particles

This paper deals with the theoretical investigation of the combined effect of magnetic field dependent (MFD) viscosity and rotation on ferroconvection saturating a porous medium in the presence of dust particles subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis has been carried out to study the onset of ferroconvection. The cases of stationary convection and oscillatory modes have been discussed. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of overstability.     

Effect of temperature dependent viscosity on the onset of Bénard–Marangoni ferroconvection

The onset of coupled Bénard–Marangoni convection in a horizontal layer of ferrofluid with viscosity depending exponentially on temperature is investigated. The lower rigid and the upper free boundaries are assumed to be insulated to temperature perturbations and the free boundary at which the surface tension effects are accounted for is assumed to be non-deformable. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by a regular perturbation technique with a wave number as a perturbation parameter. The analytical and numerically computed results are found to be in concurrence. The combined effect of magnetic number M₁ and the viscosity parameter B is to reinforce together and to hasten the onset of Bénard–Marangoni ferroconvection compared to their presence in isolation. Nonetheless, the effect of increasing B also shows initially some stabilizing effect on the system depending on the strength of magnetic and buoyancy forces. In addition, the nonlinearity of fluid magnetization M₃ is found to have no influence on the criterion for the onset of Bénard–Marangoni ferroconvection.     

Effects of MFD viscosity and LTNE on the onset of ferromagnetic convection in a porous medium

The combined effect of magnetic field dependent (MFD) viscosity and a local thermal non-equilibrium (LTNE) on the criterion for the onset of ferromagnetic convection in a ferrofluid saturated horizontal porous layer heated from below in the presence of a uniform vertical magnetic field is studied analytically using linear stability theory. A modified Darcy equation is used to describe the flow in the porous medium and a two-field model for temperature each representing the solid as well as fluid phases separately is used for energy equation. It is demonstrated that the principle of exchange of stability is valid. The results indicate that the onset of ferromagnetic convection is delayed with an increase in the MFD viscosity parameter but shows no influence on the critical wave number. Moreover, the system is found to be more stable when the magnetic forces alone are present. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H_t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic and LTNE parameters on the stability characteristics of the system is also discussed.     

The onset of Brinkman ferroconvection using a thermal non-equilibrium model

The onset of ferroconvection in a horizontal layer of ferrofluid saturated Brinkman porous medium heated uniformly from below in the presence of a uniform vertical magnetic field is investigated when the solid and fluid phases of the porous medium are in local thermal non-equilibrium (LTNE) using linear stability theory. The modified Brinkman–Forchheimer-extended Darcy equation is employed to describe the flow in the porous medium and a two-field model for energy equation each representing the solid and fluid phases separately is used. It is established that the principle of exchange of stability is valid. The authenticity of LTNE model over LTE model and also the ferromagnetic effects on the stability of the system are discussed in detail. The system is found to be more stable when the magnetic forces alone are present. Asymptotic analysis for both small and large values of scaled inter-phase heat transfer coefficient H_t is presented and the results are found to be in good agreement with those obtained from the exact formula. The established results in the literature have been reproduced as particular cases from the present study.     

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

The onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21102     

Darcy–Benard–Marangoni convection in porous media

The onset of coupled Darcy–Benard–Marangoni convection in a liquid saturated porous layer of high permeability of practical importance is investigated by employing the Brinkman–Forchheimer– Lapwood-extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid and insulating to temperature perturbations, while the upper surface is open to atmosphere and subject to a general thermal condition. The critical eigenvalues are obtained numerically, in general, using Galerkin method. However, closed form solution is also obtained using regular perturbation technique for insulated boundaries. Besides, the eigenvalue problem is solved exactly for pure Darcy–Marangoni convection. The numerical and analytical results are found to be in excellent agreement with each other. It is observed that the effect of buoyancy is destabilizing, while an increase in the permeability parameter is to delay the onset of convection. The Biot number and the ratio of effective viscosity to fluid viscosity are found to increase the critical conditions. Some known results are recovered as special cases.     

A nonlinear stability analysis for thermoconvective magnetized ferrofluid with magnetic field dependent viscosity

The generalized energy method which gives sufficient condition for the stability is developed for convection problem in a magnetized ferrofluid with magnetic field dependent (MFD) viscosity heated from below. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Both linear and nonlinear analyses are carried out and comparison of results shows a marked difference in the stability boundaries and thus indicates that the sub-critical instabilities are possible. The effect of various parameters on the sub-critical region has also been analyzed.     

Exact analytical solution of forced convection through a porous-saturated duct

In the current paper, the effect of a magnetic field on the fully developed forced convective flow and heat transfer is studied. An exact solution is extracted when the flow in the porous medium is governed by the Brinkman–Forchheimer Extended Darcy model. First, the problem formulation is explained to obtain a new system of mathematical formulation. Then, by utilizing the properties which are imposed into the problem, the exact closed-form analytical solution of the problem is explored. Finally, the main results are illustrated to show the impact of the porous media-shaped parameter, magnetic parameter, Forchheimer number, and viscosity ratio. It should be mentioned that the asymptotic results achieved in this study were compared with the exact results and it is found that they are in good agreement.     

Effect of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in porous media with variable viscosity

This work is focused on the study of combined heat and mass transfer on double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium in the presence of a first-order chemical reaction and heat generation or absorption with variable viscosity. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the non-dimensional non-similar governing equations. The governing equations for this investigation are formulated and solved numerically using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting technique. Comparisons with previously published work on special cases of the problem are performed and found to be in excellent agreement. A parametric study illustrating the influence of chemical reaction parameter, heat generation or absorption parameter, viscosity-variation parameter, buoyancy ratio and Lewis number on the fluid velocity, temperature, concentration as well as Nusselt number and Sherwood number is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.     

Hydromagnetic convective diffusion of species in Darcy–Forchheimer porous medium with non-uniform heat source/sink and variable viscosity

In this paper we have analyzed the combined effects of magnetic field and convective diffusion of species through a non-Darcy porous medium over a vertical stretching sheet with temperature dependent viscosity and non-uniform heat source/sink. The boundary layer equations are transformed into ordinary differential equations using self-similarity transformation which are then solved numerically using fifth-order Runge–Kutta Fehlberg method with shooting technique for various values of the governing parameters. The effects of electric field parameter, non-uniform heat source/sink parameters and Schmidt number on concentration profiles are analyzed and discussed graphically. Favorable comparisons with previously published work on various special cases of the problem are obtained.     

The effects of variable viscosity and thermal conductivity on heat transfer for hydromagnetic flow over a continuous moving porous plate with Ohmic heating

A numerical study of heat transfer from boundary layer flow driven by a continuous moving porous plate is proposed. The flow with electrically fluid due to the plate in the presence of a transverse magnetic field and Ohmic heating was molded as a steady, viscous, and incompressible. Both viscosity and thermal conductivity were variable and considered only a function of temperature. Similar analysis with Chebyshev finite difference method (ChFD) was developed to solve the governing equations for momentum and energy and determine the skin-friction coefficient and heat transfer rate. As the magnetic parameter and variable viscosity parameter increase, the fluid temperature and skin-friction coefficient increase and the fluid velocity and heat transfer rate decrease. The fluid temperature increases and heat transfer rate decreases with an increasing Eckert number and thermal conductivity parameter. The skin-friction coefficient and heat transfer rate increase, whereas the fluid velocity and temperature decrease as the wall suction velocity increase.     

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

The thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.     

Fluid properties of an unsteady MHD free stream flow over a stretching sheet in presence of radiation

Variable fluid properties with thermal radiation in an unsteady magnetohydrodynamics free stream incompressible flow over a stretching sheet has been considered. The thermal diffusivity and viscosity of the fluid varies linearly with temperature. The governing partial differential equations are moulded to ordinary differential equations using time-dependent similarity variables and the stream function. RKF technique with shooting method has been implement to find the solution numerically. In the current analysis the impact of unsteadiness, magnetic field, radiative parameter, variable fluid viscosity and thermal diffusivity parameter on heat and flow behavior with the free stream parameter have been studied. Transition point observed in the velocity profiles with an change in unsteadiness parameter and the effect of magnetic field is reduced in the presence of free stram velocity. The velocity and the temperature gradient are computed on the surface and their outcomes with different parameters have been analyzed in the results shown graphically and in tabular form.     

The Couette flow of a conducting Jeffrey fluid when the walls are lined with deformable porous material

The paper deals with the flow, past a deformable porous channel bounded by finite deformable porous layer with moving rigid parallel plates. Transverse magnetic field is also applied and incorporated in the momentum equation. The coupled nonlinear equations are transformed to ordinary differential equations (ODEs) with suitable choice of similarity transformation. Further, these sets of nonlinear ODEs are solved analytically and are used to get results for the flow phenomena. The effects of the porous layer thickness and the drag on the flow phenomena are discussed graphically. It is observed that rigid velocity decreases with increasing in the drag, whereas the decrease in the deformable is noted. It is clear to see that the retards in solid displacement are shown with enhancing viscosity parameter η.     

Magnetic and buoyancy effects on melting from a vertical plate embedded in saturated porous media

The magnetic and buoyancy effects on melting processes about a vertical wall embedded in a saturated porous medium are investigated. The Forchheimer extension is considered in the flow equations, and the magnetic work is included in the energy equation. A similarity solution for the transformed governing equations is obtained, and the combined effect of magnetic field on heat transfer rate is discussed. Numerical results for the velocity and temperature profiles as well as Nusselt number have been presented. The effect of inertial forces on flow and heat transfer in porous media is analyzed. The Nusselt number was found to decrease at the solid–liquid interface as the melting parameter increases.     

Mathematical modeling of onset of convection in a porous layer with viscosity variation

IntroduthenIn a novel EOR (Enhanced Oil Recovery) method,diffusion-convechon may be employed as a drive toforce heavy, viscous oil to move. A ~ of gas isexpanded beneath oil, and gas is allowed tO diffuseupward. As oil dissolves gas, it becomes lighter.Therefore, an inverse density gradient is develOPed in oillayer gradually. When this density gradient exceeds acritical value, convechon is started. It may be concjudedthat develOPing a theory, which predictS the onset ofconvection, should…     

Effect of variable gravity field on soret driven thermosolutal convection in a porous medium

The onset of thermosolutal convection due to thermal diffusion in a fluid saturated isotropic porous layer, subject to a gravity gradient is investigated using the Galerkin technique. Results reveal that the Soret parameter affects the pattern of convection only when its magnitude is large, both in the presence and absence of variable gravity field.     

Nonlinear instability analysis of internally heated convection in a porous layer with the impact of magnetic field and variable gravity

The impact of the magnetic field, heat source, and variable gravity field on the stability of convective phenomena in a porous layer are investigated numerically. Three types of gravity variations, such as linear, parabolic, and cubic functions are considered. For linear and nonlinear theories, the method of normal modes has been employed to solve governing dimensionless equations. It is found that the onset of convection is delayed by increasing the Hartmann number and gravity variation parameter and enhancement of the internal heat source makes the system stable. The subcritical instability region increases as the Hartmann number and gravity variation parameter increase.     

Effects of non-Darcian on forced convection heat transfer over a flat plate in a porous medium-with temperature dependent viscosity

The non-Darcian effect on forced convection heat transfer over a flat plate in a porous medium is examined. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The effects of inertia forces and the distance from the leading edge of the plate on the velocity and temperature fields as well as on the skin friction and heat transfer coefficients in the boundary layer over a semi-infinite plate are studied. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the Keller box method. The velocity profiles, temperature profiles and the skin friction components on the plate are computed and discussed in detail numerically for various values of the variable viscosity parameter, the modified Reynolds number, the stream wise coordinate and the Prandtl number.